Geometric construction and the Smale–Williams attractor Each solenoid may be constructed as the intersection of a nested system of embedded solid tori in
R3. Fix a sequence of natural numbers {
ni},
ni ≥ 2. Let
T0 =
S1 ×
D be a
solid torus. For each
i ≥ 0, choose a solid torus
Ti+1 that is wrapped longitudinally
ni times inside the solid torus
Ti. Then their intersection : \Lambda=\bigcap_{i\ge 0}T_i is
homeomorphic to the solenoid constructed as the inverse limit of the system of circles with the maps determined by the sequence {
ni}. Here is a variant of this construction isolated by
Stephen Smale as an example of an
expanding attractor in the theory of smooth dynamical systems. Denote the angular coordinate on the circle
S1 by
t (it is defined mod 2π) and consider the complex coordinate
z on the two-dimensional
unit disk D. Let
f be the map of the solid torus
T =
S1 ×
D into itself given by the explicit formula : f(t,z) = \left(2t, \tfrac{1}{4}z + \tfrac{1}{2}e^{it}\right). This map is a smooth
embedding of
T into itself that preserves the
foliation by meridional disks (the constants 1/2 and 1/4 are somewhat arbitrary, but it is essential that 1/4
i is the image of
T under the
ith iteration of the map
f. This set is a one-dimensional (in the sense of
topological dimension)
attractor, and the dynamics of
f on
Λ has the following interesting properties: • meridional disks are the
stable manifolds, each of which intersects
Λ over a
Cantor set •
periodic points of
f are
dense in
Λ • the map
f is
topologically transitive on
Λ General theory of solenoids and expanding attractors, not necessarily one-dimensional, was developed by R. F. Williams and involves a projective system of infinitely many copies of a compact
branched manifold in place of the circle, together with an expanding self-
immersion.
Construction in toroidal coordinates In the
toroidal coordinates with radius R, the solenoid can be parametrized by t\in \R as\zeta = 2\pi t, \quad re^{i\theta} = \sum_{k=1}^\infty r_k e^{2\pi i \omega_k t}where \omega_k = \frac{1}{n_1 \cdots n_k}, \quad r_k = R \delta_1 \cdots \delta_k Here, \delta_k are adjustable shape-parameters, with constraint 0 . In particular, \delta = \frac{1}{2n_k} works. Let S\subset \R^3 be the solenoid constructed this way, then the topology of the solenoid is just the subset topology induced by the
Euclidean topology on \R^3. Since the parametrization is bijective, we can pullback the topology on S to \R, which makes \R itself the solenoid. This allows us to construct the inverse limit maps explicitly:g_k: \R \to S_k, \quad g_k(t) = (r, \theta, \zeta)\text{ in toroidal coordinates, where } \zeta = 2\pi t, \quad re^{i\theta} = \sum_{k=1}^k r_k e^{2\pi i \omega_k t}
Construction by symbolic dynamics Viewed as a set, the solenoid is just a Cantor-continuum of circles, wired together in a particular way. This suggests to us the construction by
symbolic dynamics, where we start with a circle as a "racetrack", and append an "odometer" to keep track of which circle we are on. Define S = S^1 \times \prod_{k=1}^\infty \Z_{n_k} as the solenoid. Next, define addition on the odometer \Z \times \prod_{k=1}^\infty \Z_{n_k} \to \prod_{k=1}^\infty \Z_{n_k}, in the same way as
p-adic numbers. Next, define addition on the solenoid +: \R \times S \to S byr + (\theta, n) = ((r + \theta \mod 1), \lfloor r + \theta \rfloor + n) The topology on the solenoid is generated by the basis containing the subsets S' \times Z'_{(m_1, ..., m_k)} , where S' is any open interval in S^1 , and Z'_{(m_1, ..., m_k)} is the set of all elements of \prod_{k=1}^\infty \Z_{n_k} starting with the initial segment (m_1, ..., m_k) . == Pathological properties ==